Closure of a subgroup in an infinite Galois extension

Let $K/F$ be a Galois extension. Let $H$ be a subgroup of $G=\mathrm{Gal}(K|F)$. Let $\mathcal{N}=\lbrace N\subseteq G\text{ }|\text{ } N= \text{Gal}(K|E)\text{ where }[E:F]<\infty \text{ and } E/F \text{ is Galois}\rbrace$.

Let $\overline{H}$ be the closure of $H$. (with respect to the Krull topology). Prove that $$\overline{H}=\cap_{N\in\mathcal{N}}HN.$$

• Is this an assigned (e.g. homework) problem? Are you asking others to prove it for you? If “no” to both questions, why don’t you tell us what ideas you’ve had and how far you’ve gotten with them? – Lubin Jul 9 '16 at 3:02
• It is an assignment problem. I only have the idea that $\overline{H}=\text{Gal}(K|L)$ where $L$ is the fixed field of $H$. – learning_math Jul 9 '16 at 11:05
• Is it true that $HN$ is open for any $N\in\mathcal{N}$ ? This is because $HN=\cup_{h\in H}hN$ and $hN$ is open for all $h\in H$. As $HN$ is a union of open sets, it is open. – learning_math Jul 9 '16 at 11:14
• Is it true that $HN$ is closed for all $N\in\mathcal{N}$ ? – learning_math Jul 9 '16 at 11:18